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Vol.13
No.9&10, September 1, 2013
Research Articles:
The local Hamiltonian problem on a line with eight
staes is QMA-complete
(pp0721-0750)
Sean
Hallgren, Daniel Nagaj, and Sandeep Narayanaswami
The Local Hamiltonian problem is the problem of estimating the least
eigenvalue of a local Hamiltonian, and is complete for the class QMA.
The 1D problem on a chain of qubits has heuristics which work well,
while the 13-state qudit case has been shown to be QMA-complete. We show
that this problem remains QMA-complete when the dimensionality of the
qudits is brought down to 8.
Entanglement distillation by
extendible maps (pp0751-0770)
Lukasz
Pankowski, Fernando G.S.L. Brandao, Michal Horodecki, and Graeme Smith
It is known that from entangled states that have positive partial
transpose it is not possible to distill maximally entangled states by
local operations and classical communication (LOCC). A long-standing
open question is whether maximally entangled states can be distilled
from every state with a non-positive partial transpose. In this paper we
study a possible approach to the question consisting of enlarging the
class of operations allowed. Namely, instead of LOCC operations we
consider $k$-extendible operations, defined as maps whose Choi-Jamio\l{}kowski
state is $k$-extendible. We find that this class is unexpectedly
powerful - e.g. it is capable of distilling EPR pairs even from
completely product states. We also perform numerical studies of
distillation of Werner states by those maps, which show that if we raise
the extension index $k$ simultaneously with the number of copies of the
state, then the class of $k$-extendible operations are not that powerful
anymore and provide a better approximation to the set of LOCC
operations.
Reversible logic synthesis by
quantum rotation gates (pp0771-0792)
Afshin
Abdollahi, Mehdi Saeedi, and Massoud Pedram
A rotation-based synthesis framework for reversible logic is
proposed. We develop a canonical representation based on binary decision
diagrams and introduce operators to manipulate the developed
representation model. Furthermore, a recursive functional
bi-decomposition approach is proposed to automatically synthesize a
given function. While Boolean reversible logic is particularly
addressed, our framework constructs intermediate quantum states that may
be in superposition, hence we combine techniques from reversible Boolean
logic and quantum computation. {The proposed approach results in
quadratic gate count for multiple-control Toffoli gates without ancillae,
linear depth for quantum carry-ripple adder, and $O(n\log^2 n)$ size for
quantum multiplexer.
Upper bounds on the rate of low
density stabilizer codes for the quantum erasure channel (pp0793-0826)
Nicolas
Delfosse and Gilles Zemor
Using combinatorial arguments, we determine an upper bound on
achievable rates of stabilizer codes used over the quantum erasure
channel. This allows us to recover the no-cloning bound on the capacity
of the quantum erasure channel, $R \leq 1-2p$, for stabilizer codes: we
also derive an improved upper bound of the form $R \leq 1-2p-D(p)$ with
a function $D(p)$ that stays positive for $0<p<1/2$ and for any family
of stabilizer codes whose generators have weights bounded from above by
a constant -- low density stabilizer codes. We obtain an application to
percolation theory for a family of self-dual tilings of the hyperbolic
plane. We associate a family of low density stabilizer codes with
appropriate finite quotients of these tilings. We then relate the
probability of percolation to the probability of a decoding error for
these codes on the quantum erasure channel. The application of our upper
bound on achievable rates of low density stabilizer codes gives rise to
an upper bound on the critical probability for these tilings.
A study of BB84 protocol in a
device-independent scenario: from the view of entanglement distillation
(pp0827-0832)
Zhen-Qiang
Yin, Wei Chen, Shuang Wang, Hong-Wei Li, Guang-Can Guo, and Zheng-Fu Han
For the past few years, the security of practical quantum key
distribution systems has attracted a lot of attention.
Device-independent quantum key distribution was proposed to design a
real-life secure quantum key distribution system with imperfect and
untrusted quantum devices. In this paper, we analyzed the security of
BB84 protocol in a device-independent scenario based on the entanglement
distillation method. Since most of the reported loopholes are in
receivers of quantum key distribution systems, we focus on condition
that the transmitter of the system is perfectly coincident with the
requirement of the BB84 protocol, while the receiver can be controlled
by eavesdropper. Finally, the lower bound of the final secret-key rate
was proposed and we explained why the secure-key rate is similar to the
well-known result for the original entanglement distillation protocol.
Pseudo-telepathy games using graph
states (pp0833-0845)
Anurag Anshu
and Mehdi Mhalla
We define a family of pseudo-telepathy games using graph states that
extends the Mermin games. This family also contains a game used to
define a quantum probability distribution that cannot be simulated by
any number of nonlocal boxes. We extend this result, proving that the
probability distribution obtained by the Paley graph state on 13
vertices (each vertex corresponds to a player) cannot be simulated by
any number of 4-partite nonlocal boxes and that the Paley graph states
on $k^{2}2^{2k-2}$ vertices provide a probability distribution that
cannot be simulated by $k$-partite nonlocal boxes, for any $k$.
Full characterization of quantum
correlated equilibria (pp0846-0860)
Zhaohui
Wei and Shengyu Zhang
Quantum game theory aims to study interactions of people (or other
agents) using quantum devices with possibly conflicting interests.
Recently Zhang studied some quantitative questions in general quantum
strategic games of growing sizes~\cite{Zha12}. However, a fundamental
question not addressed there is the characterization of quantum
correlated equilibria (QCE). In this paper, we answer this question by
giving a sufficient and necessary condition for an arbitrary state
$\rho$ being a QCE. In addition, when the condition fails to hold for
some player $i$, we give an explicit positive-operator valued
measurement (POVM) for that player to achieve a strictly positive gain
of payoff. Finally, we give some upper bounds for the maximum gain by
playing quantum strategies over classical ones, and the bounds are tight
for some games.
Security of plug-and-play QKD arrangements with
finite resources (pp0861-0879)
Pedro J. Salas
The security of a passive
plug-and-play QKD arrangement in the case of finite (resources) key
lengths is analysed. It is assumed that the eavesdropper has full access
to the channel so an unknown and untrusted source is assumed. To take
into account the security of the BB84 protocol under collective attacks
within the framework of quantum adversaries, a full treatment provides
the well-known equations for the secure key rate. A numerical simulation
keeping a minimum number of initial parameters constant as the total
error sought and the number of pulses is carried out. The remaining
parameters are optimized to produce the maximum secure key rate. Two
main strategies are addressed: with and without two-decoy-states
including the optimization of signal to decoy relationship.
Optimal asymmetric quantum cloning for quantum
information and computation (pp0880-0900)
Alastair Kay,
Ravishankar Ramanathan, and Dagomir Kaszlikowski
While the no-cloning theorem, which forbids the perfect copying of
quantum states, is well-known as one of the defining features of quantum
mechanics, the question of how well the theory allows a state to be
cloned is yet to be completely solved. In this paper, rigorous solutions
to the problem of $M\rightarrow N$ asymmetric cloning of qudits are
obtained in a number of interesting cases. The central result is the
solution to the $1 \rightarrow N$ universal asymmetric qudit cloning
problem for which the exact trade-off in the fidelities of the clones
for every $N$ and $d$ is derived. Analogous results are proven for
qubits when $M=N-1$. We also consider state-dependent $1 \rightarrow N$
qubit cloning, providing a general parametrization in terms of a
Heisenberg star Hamiltonian. In all instances, we determine the
feasibility of implementing the cloning economically, i.e., without an
ancilla, and determine the dimension of the ancilla when an economic
implementation is not possible.
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